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\begin{document} \huge

\begin{center}
{Mathematical Physics --- PHZ 3113}\\\vspace {2mm} 
{\bf Levi-Cevita Tensor 1} \\ (January 11, 2013) \\
\end{center}
\setcounter{equation}{0}

\rightline{Group \#~~~~~~~~}

\leftline{Participating students (print):} \vskip 60pt

\centerline{------------------------------------------------------}

\bigskip

\begin{enumerate}
\item Use binary numbers 0, 1 and write down the numbers 0 to 3.
Add one more column in which you substitute $0\to 1$, $1\to 2$ and 
one last column in which you count the decimal numbers from 1 to 4.

0 00 11 1\\ 1 01 12 2\\ 2 10 21 3 \\ 3 11 22 4

\item Use numbers with base 3 and symbols 0, 1, 2 to write down 
the numbers 0 to 26. Add one more column in which you substitute 
$n\to n+1$ for $n=0,\,1,\,2$ and one last column in which you count 
the decimal numbers from 1 to 27. %  \eject

$\left.\right.$
 0 000 111 ~1\\ $\left.\right.$ 1 001 112 ~2\\ $\left.\right.$ 
 2 002 113 ~3\\ $\left.\right.$ 3 010 121 ~4\\ $\left.\right.$ 
 4 011 122 ~5\\ $\left.\right.$ 5 012 123 ~6\\ $\left.\right.$ 
 6 020 131 ~7\\ $\left.\right.$ 7 021 132 ~8\\ $\left.\right.$ 
 8 022 133 ~9\\ $\left.\right.$ 9 100 211 10\\  10 101 212 11\\ 
11 102 213 12\\ 12 110 221 13\\ 13 111 222 14\\ 14 112 223 15\\ 
15 120 231 16\\ 16 121 232 17\\ 17 122 233 18\\ 18 200 311 19\\ 
19 201 312 20\\ 20 202 313 21\\ 21 210 321 22\\ 22 211 322 23\\ 
23 212 323 24\\ 24 220 331 25\\ 25 221 332 26\\ 26 222 333 27

\item Write down the {\bf permutations} of $1\,2$.
How many permutations of $1\,2$ are there? 
\begin{equation} \label{p12}
 1\,2~~~2\,1~~~{\rm there\ are\ 2\ permuations.}
\end{equation}

\item Write down the transposition of $1\,2$. 
\begin{equation} 
 2\,1
\end{equation} 

\item Use the permutations of (\ref{p12}) to write down the 
permutations of $1\,2\,3$ by starting in each case with 3 on 
the right and transposing the number 3 with its left neighbor
until this is no longer possible. How many permutations of 
$1\,2\,3$ are there?
\begin{eqnarray} \label{p123}
 1\,2\,3~~~1\,3\,2~~~3\,1\,2\\ \nonumber
 2\,1\,3~~~2\,3\,1~~~3\,2\,1
\end{eqnarray} 
There are 6 permutations.

\item Along the same lines: Use the permutations of (\ref{p123}) to 
write down the permutations of $1\,2\,3\,4$. How many permutations 
of $1\,2\,3\,4$ are there?
\begin{eqnarray} \label{p1234}
 1\,2\,3\,4~~~1\,2\,4\,3~~~1\,4\,2\,3~~~4\,1\,2\,3~~~\\ \nonumber
 1\,3\,2\,4~~~1\,3\,4\,2~~~1\,4\,3\,2~~~4\,1\,3\,2~~~\\ \nonumber
 3\,1\,2\,4~~~3\,1\,4\,2~~~3\,4\,1\,2~~~4\,3\,1\,2~~~\\ \nonumber
 2\,1\,3\,4~~~2\,1\,4\,3~~~2\,4\,1\,3~~~4\,2\,1\,3~~~\\ \nonumber
 2\,3\,1\,4~~~2\,3\,4\,1~~~2\,4\,3\,1~~~4\,2\,3\,1~~~\\ \nonumber
 3\,2\,1\,4~~~3\,2\,4\,1~~~3\,4\,2\,1~~~4\,3\,2\,1~~~ \nonumber
\end{eqnarray} 
There are $24=4!$ permutations.

\item Proof that there are $n!$ permutions $\pi_1,\dots,\pi_n$
of $1,\dots,n$. \bigskip

Thee proof is by induction. Assume that there are $(n-1)!$ permutations 
for $$\pi_1,\dots,\pi_{n-1}$$ as we have verified up to $n-1=4$. From 
each of these permuations we get $n$ new permuations by starting with 
the number $n$ on the right side and transposing it in $(n-1)$ steps to 
the left side:
\begin{eqnarray} \nonumber
  \pi_1,&\dots&,\pi_{n-1},n \\ \nonumber
  \pi_1,&\dots&,n,\pi_{n-1} \\ \nonumber
        &\dots&             \\ \nonumber
  \pi_1,n,&\dots&,\pi_{n-1} \\ \nonumber
  n,\pi_1,&\dots&,\pi_{n-1} 
\end{eqnarray} 
Hence there are $$(n-1)!\,n\ =\ n!$$ permuations of the numbers
$1,\dots,n$. \eject

\item With 
\begin{eqnarray} \nonumber
  i_1&=&1,\dots,n\,, \\ \nonumber
  i_2&=&1,\dots,n\,, \\ \nonumber
     &~&  \dots      \\ \nonumber
  i_n&=&1,\dots,n
\end{eqnarray} 
the definition of the {\bf Levi-Cevita tensor} is
\begin{eqnarray} \nonumber
  \epsilon_{i_1,\dots,i_n} &=& \cases{
  +1~{\rm for}~i_1,\dots,i_n~{\rm even~permutation,} \cr
  -1~{\rm for}~i_1,\dots,i_n~{\rm odd~permutation,} \cr
   0~{\rm for}~i_1,\dots,i_n~{\rm no~permutation.} }\\ &~& 
\end{eqnarray}
A permutation $i_1,\dots,i_n$ is even, when it is generated
by an even number of transpositions of $1,\dots,n$ and odd,
when it is generated by an odd number of transpositions of 
$1,\dots,n$.  

How many non-zero elements are there?  
\begin{eqnarray} n! \end{eqnarray}
How many positive elements are there? 
\begin{eqnarray} n!/2 \end{eqnarray} % \eject

\item Write down all elements of the Levi-Cevita tensor for $n=2$.
\begin{eqnarray} \nonumber 
\epsilon_{11}= 0 &~& \epsilon_{12}=+1 \\
\epsilon_{21}=-1 &~& \epsilon_{22}= 0
\end{eqnarray} % \eject

\item Write down all elements of the Levi-Cevita tensor for $n=3$.
\begin{eqnarray} 
 1.~~\epsilon_{111} &=& 0 \\
 2.~~\epsilon_{112} &=& 0 \nonumber \\
 3.~~\epsilon_{113} &=& 0 \nonumber \\
 4.~~\epsilon_{121} &=& 0 \nonumber \\
 5.~~\epsilon_{122} &=& 0 \nonumber \\
 6.~~\epsilon_{123} &=& +1 \nonumber \\
 7.~~\epsilon_{131} &=& 0 \nonumber \\
 8.~~\epsilon_{132} &=& -1 \nonumber \\
 9.~~\epsilon_{133} &=& 0 \nonumber \\
10.~~\epsilon_{211} &=& 0 \nonumber \\
11.~~\epsilon_{212} &=& 0 \nonumber \\
12.~~\epsilon_{213} &=& -1 \nonumber \\
13.~~\epsilon_{221} &=& 0 \nonumber \\
14.~~\epsilon_{222} &=& 0 \nonumber \\
15.~~\epsilon_{223} &=& 0 \nonumber \\
16.~~\epsilon_{231} &=& +1 \nonumber \\
17.~~\epsilon_{232} &=& 0 \nonumber \\
18.~~\epsilon_{233} &=& 0 \nonumber \\
19.~~\epsilon_{311} &=& 0 \nonumber \\
20.~~\epsilon_{312} &=& +1 \nonumber \\
21.~~\epsilon_{313} &=& 0 \nonumber \\
22.~~\epsilon_{321} &=& -1 \nonumber \\
23.~~\epsilon_{322} &=& 0 \nonumber \\
24.~~\epsilon_{323} &=& 0 \nonumber \\
25.~~\epsilon_{331} &=& 0 \nonumber \\
26.~~\epsilon_{332} &=& 0 \nonumber \\
27.~~\epsilon_{333} &=& 0 \nonumber 
\end{eqnarray}

\item Write down all non-zero elements of the Levi-Cevita tensor for 
$n=4$. 
\begin{eqnarray} 
 1.~~\epsilon_{1234} &=& +1 \nonumber \\ 
 2.~~\epsilon_{1243} &=& -1 \nonumber \\
 3.~~\epsilon_{1423} &=& +1 \nonumber \\
 4.~~\epsilon_{4123} &=& -1 \nonumber \\
 ~~\nonumber \\
 5.~~ \epsilon_{1324} &=& -1 \nonumber \\
 6.~~\epsilon_{1342} &=& +1 \nonumber \\
 7.~~\epsilon_{1432} &=& -1 \nonumber \\
 8.~~\epsilon_{4132} &=& +1 \nonumber \\
 ~~\nonumber \\
 9.~~\epsilon_{3124} &=& +1 \nonumber \\
10.~~\epsilon_{3142} &=& -1 \nonumber \\
11.~~\epsilon_{3412} &=& +1 \nonumber \\
12.~~\epsilon_{4312} &=& -1 \nonumber \\
 ~~\nonumber \\
13.~~ \epsilon_{2134} &=& -1 \nonumber \\
14.~~\epsilon_{2143} &=& +1 \nonumber \\
15.~~\epsilon_{2413} &=& -1 \nonumber \\
16.~~\epsilon_{4213} &=& +1 \nonumber \\
 ~~\nonumber \\
17.~~ \epsilon_{2314} &=& +1 \nonumber \\
18.~~\epsilon_{2341} &=& -1 \nonumber \\
19.~~\epsilon_{2431} &=& +1 \nonumber \\
20.~~\epsilon_{4213} &=& -1 \nonumber \\
 ~~\nonumber \\
21.~~ \epsilon_{3214} &=& -1 \nonumber \\
22.~~\epsilon_{3241} &=& +1 \nonumber \\
23.~~\epsilon_{3241} &=& -1 \nonumber \\
24.~~\epsilon_{4321} &=& +1 \nonumber 
\end{eqnarray}

\end{enumerate}

\end{document}

